On the completeness of the system in

Let E = ⋃ n = 1 ∞ I n be the union of infinitely many disjoint closed intervals where I n = [ a n , b n ] , 0 < a 1 < b 1 < a 2 < b 2 < ⋯ < b n < ⋯ , and dist ( 0 , I n ) → ∞ . Let α ( t ) be a nonnegative function and { λ n } n = 1 ∞ be a sequence of distinct complex numbers. Motivated by the work of Guantie Deng, in this paper, a theorem on the completeness of the system { t λ n } in C 0 ( E ) is obtained where C 0 ( E ) is the weighted Banach space which consists of complex functions continuous on E with f ( t ) e − α ( t ) vanishing at infinity.